Heyting arithmetic

In mathematical logic, Heyting arithmetic (sometimes abbreviated HA) is an axiomatization of arithmetic in accordance with the philosophy of intuitionism (Troelstra 1973:18). It is named after Arend Heyting, who first proposed it.

Heyting arithmetic adopts the axioms of Peano arithmetic (PA), but uses intuitionistic logic as its rules of inference. In particular, the law of the excluded middle does not hold in general, though the induction axiom can be used to prove many specific cases. For instance, one can prove that ∀ x, y ∈ N : x = y ∨ x ≠ y is a theorem (any two natural numbers are either equal to each other, or not equal to each other). In fact, since "=" is the only predicate symbol in Heyting arithmetic, it then follows that, for any quantifier-free formula p, ∀ x, y, z, … ∈ N : p ∨ ¬p is a theorem (where x, y, z… are the free variables in p).

Kurt Gödel studied the relationship between Heyting arithmetic and Peano arithmetic. He used the Gödel–Gentzen negative translation to prove in 1933 that if HA is consistent, then PA is also consistent.

Heyting arithmetic should not be confused with Heyting algebras, which are the intuitionistic analogue of Boolean algebras.

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